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Math Help - Metric space question

  1. #1
    pkr
    pkr is offline
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    Metric space question

    Can someone help with this please, i'm stuck;

    X = [0,∞) and d(x,y) = │x-y│/1+│x-y│

    Show that it's a Metric space; I've done the first two conditions but stuck proving the triangle inequality...

    Then it says describe B1(0) the ball of centre 0 and radius 1...surely there's not much to say?
    Last edited by mr fantastic; February 11th 2009 at 01:37 PM. Reason: Moved to new thread, edited accordingly.
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  2. #2
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    \frac{{\left| {x - y} \right|}}{{1 + \left| {x - y} \right| + \left| {y - z} \right|}} \leqslant \frac{{\left| {x - y} \right|}}{{1 + \left| {x - y} \right|}} = d(x,y)

    \frac{{\left| {y - z} \right|}}{{1 + \left| {x - y} \right| + \left| {y - z} \right|}} \leqslant \frac{{\left| {y - z} \right|}}{{1 + \left| {y - z} \right|}} = d(y,z)

    d(x,z) = \frac{{\left| {x - z} \right|}}<br />
{{1 + \left| {x - y} \right| + \left| {y - z} \right|}} \leqslant \frac{{\left| {x - y} \right| + \left| {y - z} \right|}}<br />
{{1 + \left| {x - y} \right| + \left| {y - z} \right|}}
    d(x,z) \leqslant \frac{{\left| {x - y} \right| + \left| {y - z} \right|}}<br />
{{1 + \left| {x - y} \right| + \left| {y - z} \right|}} \leqslant \frac{{\left| {x - y} \right|}}<br />
{{1 + \left| {x - y} \right|}} + \frac{{\left| {y - z} \right|}}<br />
{{1 + \left| {y - z} \right|}} = d(x,y) + d(y,z)
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